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User:Ilya Gutkovskiy

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Location: Tula, Russia.

Born in 1982.

Graduated from Tula State University (Faculty of Technology) and Tula State Pedagogical University (Faculty of Psychology).

I work in the field of metal processing industry.

I am interested in elementary and analytic number theory. In addition to mathematics I am fond of poetry. My books


OEIS sequences which I submitted

OEIS sequences which I submitted and/or edited


Contents

Generalization of generating functions

The ordinary generating function for the alternating sum of k-gonal numbers

\frac{-x(1-(k-3)x)}{(1-x)(1+x)^{3}}

A266088

The ordinary generating function for the alternating sum of centered k-gonal numbers

\frac{1-(k-2)x+x^{2}}{(1-x)(1+x)^{3}}

A270693

The ordinary generating function for the alternating sum of k-gonal pyramidal numbers

\frac{-x(1-(k-3)x)}{(1-x)(1+x)^{4}}

A266677

The ordinary generating function for the alternating sum of centered k-gonal pyramidal numbers

\frac{-x(1-(k-2)x+x^{2})}{(1-x)(1+x)^{4}}

A270694

The ordinary generating function for the first bisection of k-gonal numbers

\frac{kx+(3k-8)x^{2}}{(1-x)^{3}}

A270704

The ordinary generating function for the first trisection of k-gonal numbers

\frac{3x(k-1+(2k-5)x)}{(1-x)^{3}}

A268351

The ordinary generating function for the squares of k-gonal numbers

\frac{x(1+(k^{2}-5)x+(4k^{2}-18k+19)x^{2}+(k-3)^{2}x^{3})}{(1-x)^{5}}

A100255

The ordinary generating function for the convolution of nonzero k-gonal numbers and nonzero h-gonal numbers

\frac{(1+(k-3)x)(1+(h-3)x)}{(1-x)^{6}}

A271663

The ordinary generating function for the convolution of nonzero k-gonal numbers with themselves

\frac{(1+(k-3)x)^{2}}{(1-x)^{6}}

A271662

The ordinary generating function for the convolution of nonzero k-gonal numbers and nonzero triangular numbers

\frac{1+(k-3)x)}{(1-x)^{6}}

A271567

The ordinary generating function for the generalized k-gonal numbers

\frac{x(1+(k-4)x+x^{2})}{(1-x)^{3}(1+x)^2}

A277082

The ordinary generating function for the Sum_{k = 0..n} m^k

\frac{1}{(1-mx)(1-x)}

A269025

The ordinary generating function for the Sum_{k = 0..n} (-1)^k*m^k

\frac{1}{1+(m-1)x-mx^{2}}

A268413

The ordinary generating function for the Sum_{k=0..n} floor(k/m)

\frac{x^{m}}{(1-x^{m})(1-x)^{2}}

A269445

The ordinary generating function for the sums of m consecutive squares of nonnegative integers

\frac{m(1-2x+13x^{2}+2m^{2}(1-2x+x^{2})-3m(1-4x+3x^{2}))}{6(1-x)^3}

A276026

The ordinary generating function for the number of ways of writing n as a sum of k squares

\vartheta _{3}(0,q)^{k}=1+2kq+2(k-1)q^{2}+\frac{4}{3}(k-2)(k-1)kq^{3}+...

A276285

The ordinary generating function for the values of quadratic polynomial p*n^2 + q*n + k

\frac{k+(p+q-2k)x+(p-q+k)x^{2}}{(1-x)^3}

A270710

The ordinary generating function for the values of cubic polynomial p*n^3 + q*n^2 + k*n + m

\frac{m+(p+q+k-3m)x+(4p-2k+3m)x^{2}+(p-q+k-m)x^{3}}{(1-x)^{4}}

A268644

The ordinary generating function for the values of quartic polynomial p*n^4 + q*n^3 + k*n^2 + m*n + r

\frac{(r+(p+q+k+m-4r)x+(11p+3q-k-3m+6r)x^{2}+(11p-3q-k+3m-4r)x^{3}+(p-q+k-m+r)x^{4}}{(1-x)^{5}}

A269792

The ordinary generating function for the values of quintic polynomial b*n^5 + p*n^4 + q*n^3 + k*n^2 + m*n + r

\frac{r+(b+p+q+k+m-5r)x+(13b+5p+q-k-2m+5r)2x^{2}+(33b-3q+3m-5r)2x^{3}+(26b-10p+2q+2k-4m+5r)x^{4}+(b-p+q-k+m-r)x^{5}}{(1-x)^{6}}

A125083

The ordinary generating function for the characteristic function of the multiples of k

\frac{1}{1-x^{k}}

A267142

The ordinary generating function for the continued fraction expansion of phi^(2*k + 1), where phi = (1 + sqrt(5))/2), k = 1, 2, 3,...

\frac{\left \lfloor \varphi ^{2k+1} \right \rfloor}{1-x}

A267319

The ordinary generating function for the continued fraction expansion of phi^(2*k), where phi = (1 + sqrt(5))/2), k = 1, 2, 3,...

\frac{\left \lfloor \varphi ^{2k} \right \rfloor+x-x^{2}}{1-x^2}

A267319

The ordinary generating function for the continued fraction expansion of exp(1/k), with k = 1, 2, 3....

\frac{1+(k-1)x+x^{2}-(k+1)x^{3}+7x^{4}-x^{5}}{(1-x^{3})^{2}}

A267318

The ordinary generating function for the Fibonacci(k*n)

\frac{\left (  \varphi^{k}-\left ( -\frac{1}{\varphi } \right )^{k}\right )x}{\sqrt{5}\left ( 1-\left ( \varphi ^{k}+\left ( -\frac{1}{\varphi } \right )^{k} \right )x+(-1)^{k}x^{2} \right )}

A269500

The ordinary generating function for the Sum_{k = 0..n} (k mod m)

\frac{\sum_{k=1}^{m-1}kx^{k}}{(1-x^{m})(1-x)}

A268291

The ordinary generating function for the recurrence relation b(n) = k^n - b(n-1), with n>0 and b(0)=0

\frac{kx}{(1+x)(1-kx)}

A271427

The ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - m*n, with n>0 and b(0)=1

\frac{1-(m+2)x+x^{2}}{(1-x)^{2}(1-kx)}

A268414

The ordinary generating function for the recurrence relation b(n) = r*b(n - 1) + s*b(n - 2), with n>1 and b(0)=k, b(1)=m

\frac{k-(kr-m)x}{1-rx-sx^{2}}

A268409

The ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - m*b(n - 2), with n>1 and b(0)=0, b(1)=1

\frac{x}{1-kx+mx^{2}}

A268344

The ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - b(n - 2), with n>1 and b(0)=1, b(1)=1

\frac{1-(k-1)x}{1-kx+x^{2}}

A269028

The ordinary generating function for the recurrence relation b(n) = 2*b(n - 2) - b(n - 1), with n>1 and b(0)=k, b(1)=m

\frac{k+(k+m)x}{1+x-2x^{2}}

A268741

The ordinary generating function for the recurrence relation b(n) = b(n - 1) + 2*b(n - 2) + 3*b(n - 3) + 4*b(n - 4) + ... + k*b(n - k), with n > k - 1 and initial values b(i-1) = i for i = 1..k

\frac{\sum_{m=0}^{k-1}\frac{(-m^{3}-3m^{2}+4m+6)x^{m}}{6}}{1-\sum_{m=1}^{k}mx^{m}}

A268349

The ordinary generating function for the recurrence relation b(n) = b(n - 1) + b(n - 2) + b(n - 3), with n>2 and b(0)=k, b(1)=m, b(2)=q

\frac{k+(m-k)x+(q-m-k)x^{2}}{1-x-x^{2}-x^{3}}

A268410

The ordinary generating function for the recurrence relation b(n) = floor(phi^(2*k)*b(n - 1)), with n>0, b(0)=1 and k = 0,1,2,3, ...

\frac{1-x}{1-\left ( \varphi ^{2k}+(-\varphi )^{-2k} \right )x+x^{2}}

A278475

The ordinary generating function for the recurrence relation b(n) = floor(phi^(2*k+1)*b(n - 1)), with n>0, b(0)=1 and k = 0,1,2,3, ...

\frac{1-x-x^{2}}{(1-x)(1-\left ( \varphi ^{2k+1}+(-\varphi )^{-2k-1} \right )x-x^{2})}

A278475

The ordinary generating function for the recurrence relation b(n) = floor((1 + sqrt(2))^(2*k)*b(n - 1)), with n>0, b(0)=1 and k = 1,2,3,4, ...

\frac{1-x}{1-\left [  (1+\sqrt{2})^{2k}\right ]x+x^{2}}

A278476

The ordinary generating function for the recurrence relation b(n) = floor((1 + sqrt(2))^(2*k+1)*b(n - 1)), with n>0, b(0)=1 and k = 0,1,2,3, ...

\frac{1-x-x^2}{(1-x)(1-\left [  (1+\sqrt{2})^{2k+1}\right ]x-x^{2})}

A278476

The ordinary generating function for the integers repeated k times

\frac{x^{k}}{(1-x)(1-x^k)}

A004526

The ordinary generating function for the partial sums of numbers that are repdigits in base k (for k > 1)

\frac{\sum_{m=1}^{k-1}mx^{m}}{(1-x)(1-x^{k-1})(1-kx^{k-1})}

A277209

The ordinary generating function for the binomial coefficients C(n,k)

\frac{x^k}{(1-x)^{(k+1)}}

A017764

The ordinary generation function for the Gaussian binomial coefficients [n,k]_q

\frac{x^{k}}{\prod_{m=0}^{k}(1-q^{m}x)}

A275944

The ordinary generating function for the transformation of the Wonderful Demlo numbers

\frac{kx(1+10x)}{1-111x+1110x^{2}-1000x^{3}}

A271528

The ordinary generating function for the sequences of the form k^n + m

\frac{1+m-(1+km)x}{(1-x)(1-kx)}

A271527

The ordinary generating function for the sequences of the form k*(n + 1)*(n - 1 + k)/2

\frac{\frac{k(k-1)}{2}+\left ( \frac{k(3-k)}{2} \right )x}{(1-x)^{3}}

A269457

The ordinary generating function for the number of partitions of n into parts congruent to 1 mod m (for m>0)

\prod_{k=0}^{\infty }\frac{1}{1-x^{mk+1}}

A277090

The ordinary generating function for the number of partitions of n into parts larger than 1 and congruent to 1 mod m (for m>0)

\prod_{k=1}^{\infty }\frac{1}{1-x^{mk+1}}

A277210

The ordinary generating function for the surface area of the n-dimensional sphere of radius r

2x\left ( 1+\pi \exp (\pi r^{2}x^{2})rx+2\sqrt{\pi}\exp (\pi r^{2}x^{2}) rx\int_{0}^{\sqrt{\pi}rx} \exp (-t^{2})dt\right )

A072478

The sum of reciprocals of Catalan numbers (with even indices, with odd indices)

\sum_{k=0}^{\infty }\frac{2k+1}{\binom{4k}{2k}}=\frac{2\pi }{9\sqrt{3}}-\frac{4(3\sqrt{5}\ln \varphi -40)}{125}

A276483


\sum_{k=0}^{\infty }\frac{2k+2}{\binom{4k+2}{2k+1}}=\frac{2\pi }{9\sqrt{3}}+\frac{6(2\sqrt{5}\ln \varphi +15)}{125}

A276484

Double hyperfactorial

H_2(n)=\begin{cases}
n^{n}\cdot (n-2)^{n-2}\cdot ...\cdot 5^{5}\cdot 3^{3}\cdot 1^{1},n>0,n\Rightarrow odd\\ 
n^{n}\cdot (n-2)^{n-2}\cdot ...\cdot 6^{6}\cdot 4^{4}\cdot 2^{2},n>0,n\Rightarrow even\\ 
0,n=0 
\end{cases}

H_2(n)=\prod_{k=0}^{\left \lfloor \frac{(n-1)}{2} \right \rfloor}(n-2k)^{n-2k}

H_2(n)=\frac{1}{H_2(n-1)}\sqrt{\frac{H_2(2n)}{2^{n(n+1)})}}

A271385

Polynomials

Polynomials T_n(x) = -((-1)^n*2^(-n-1)*cos(Pi*sqrt(8*x+1)/2)*Gamma(n-sqrt(8*x+1)/2+3/2)*Gamma(n+sqrt(8*x+1)/2+3/2))/Pi

File:T_n(x).gif

T_{n}(x)=\prod_{k=0}^{n}\left ( x-\sum_{m=0}^{k} m\right )

T_{n}(x)=0\Rightarrow x=0+1+2+3+...=\frac{k(k+1)}{2},k\leqslant n

A271386

Polynomials Q_n(x) = 2^(-n)*((x+sqrt(x*(x+6)-3)+1)^n-(x-sqrt(x*(x+6)-3)+1)^n)/sqrt(x*(x+6)-3)

File:Q_n(x).gif‎

G(x,t)=\frac{t}{1-(x+1)t-(x-1)t^{2}}=t+(x+1)t^{2}+x(x+3)t^{3}+..

Q_n(x)=(x+1)Q_{n-1}(x)+(x-1)Q_{n-2}(x)\Rightarrow  Q_0(x)=0,Q_1(x)=1

A271451

Polynomials C_n(x) = Sum_(k=0..n} (2*k)!*(x - 1)^(n-k)/((k + 1)!*k!)

File:C_n(x).gif

G(x,t)=\frac{1-\sqrt{1-4t}}{2t(1+t-xt)}=1+xt+(x^{2}-x+2)t^{2}+(x^{3} - 2x^{2} + 3x + 3)t^{3}+...

C_{n}(x)=(x-1)C_{n-1}+C_{n}(1)\Rightarrow  C_{0}(x)=1

C_{n}(1)=\frac{(2n)!}{(n+1)!n!}

C_{n}(2)=\sum_{m=0}^{n}C_{m}(1)

A271453

Conjectures

Every number > 15 can be represented as a sum of 3 semiprimes.

A282135

File:Number_of_ways_to_write_n_as_an_ordered_sum_of_three_semiprimes.jpg



Every number is the sum of at most 6 square pyramidal numbers.

Every number is the sum of at most k+2 k-gonal pyramidal numbers (except k = 5).

A282173

File:Number_of_ways_to_write_n_as_an_ordered_sum_of_6_square_pyramidal_numbers.jpg



Every number is the sum of at most 12 squares of triangular numbers (or partial sums of cubes).

A284641

File:Number_of_ways_to_write_n_as_an_ordered_sum_of_12_squares_of_triangular_numbers.jpg



Every number > 27 can be represented as a sum of 4 proper prime powers.

A282289

File:Number_of_ways_to_write_n_as_an_ordered_sum_of_4_proper_prime_powers.jpg



Every number > 8 can be represented as a sum of a proper prime power and a squarefree number > 1.

A282290

File:Number_of_ways_of_writing_n_as_a_sum_of_a_proper_prime_power_and_a_squarefree_number.jpg



Every number > 108 can be represented as a sum of a proper prime power and a nonprime squarefree number.

A287299

File: Number_of_ways_of_writing_n_as_a_sum_of_a_proper_prime_power_and_a_nonprime_squarefree_number.jpg



Every number > 10 can be represented as a sum of a prime and a nonprime squarefree number.

A282318

File:Number_of_ways_of_writing_n_as_a_sum_of_a_prime_and_a_nonprime_squarefree_number.jpg



Every number > 30 can be represented as a sum of a prime and a squarefree semiprime.

A282192

File:Number_of_ways_of_writing_n_as_a_sum_of_a_prime_and_a_squarefree_semiprime.jpg



Every number > 30 can be represented as a sum of a twin prime and a squarefree semiprime.

A283929

File:Number_of_ways_of_writing_n_as_a_sum_of_a_twin_prime_and_a_squarefree_semiprime_2.jpg



Every number > 108 can be represented as a sum of a perfect power and a squarefree semiprime.

A282947

File:Number_of_ways_of_writing_n_as_a_sum_of_a_perfect_power_and_a_squarefree_semiprime.jpg



Every number > 527 can be represented as a sum of a prime with prime subscript and a semiprime (only 18 positive integers cannot be represented as a sum of a prime with prime subscript and a semiprime).

A282355

File:Number_of_ways_of_writing_n_as_a_sum_of_a_prime_with_prime_subscript_and_a_semiprime.jpg



Every number > 51 can be represented as a sum of 2 multiplicatively perfect numbers.

A282570

File:Number_of_ways_to_write_n_as_an_ordered_sum_of_two_multiplicatively_perfect_numbers.jpg



Any sufficiently large number can be represented as a sum of 3 squarefree palindromes.

A282585

File:Number_of_ways_to_write_n_as_an_ordered_sum_of_3_squarefree_palindromes_in_base_10.jpg

Every number > 3 can be represented as a sum of 4 squarefree palindromes.

File:Number_of_ways_to_write_n_as_an_ordered_sum_of_4_squarefree_palindromes_in_base_10.jpg



Every number > 82 can be represented as a sum of 2 numbers that are the product of an even number of distinct primes (including 1).

A285796



Every number > 57 can be represented as a sum of 2 numbers that are the product of an odd number of distinct primes.

A285797



Every number > 10 can be represented as a sum of 2 numbers, one of which is the product of an even number of distinct primes (including 1) and another is the product of an odd number of distinct primes.

A286971



Every number > 1 is the sum of at most 5 сentered triangular numbers.

A282502



Every number > 1 is the sum of at most 6 centered square numbers.

Every number > 1 is the sum of at most k+2 centered k-gonal numbers.

A282504



Every number is the sum of at most k-4 generalized k-gonal numbers (for k >= 8).

A290943



Every number is the sum of at most 15 icosahedral numbers.

A282350



Every number > 23 is the sum of at most 8 squares of primes.

Every number > 131 can be represented as a sum of 13 squares of primes.

A275001



Every number > 16 is the sum of at most 4 primes of form x^2 + y^2.

A282971



Every number > 7 is the sum of at most 4 twin primes.

A283875



Every number > 3 is the sum of at most 5 partial sums of primes.

A282906



Let a_p(n) be the length of the period of the sequence k^p mod n where p is a prime, then a_p(n) = n/p if n == 0 (mod p^2) else a_p(n) = n.

A282779



Let a(n) be the sum of largest prime power factors of numbers <= n, then a(n) = O(n^2/log(n)).

A284521



Let a(n) = Sum_{k=1..n} sigma(k)/k, where sigma(k) is the sum of the divisors of k, it is assumed that the value of a(n)/n approaches Pi^2/6.

A284648



Let a(n) = n - a(floor(a(n-1)/2)) with a(0) = 0, then a(n) ~ c*n, where c = sqrt(3) - 1.

A286389



G.f.~ =\frac{\sum_{k=0}^{\infty } \left \lfloor  \varphi^2 (k+1) \right \rfloor x^k}{\sum_{k=0}^{\infty} \left \lfloor  \varphi (k+1) \right \rfloor x^k}=1+\frac{1}{1+\frac{x}{1+\frac{x}{1+\frac{x^2}{1 +... \frac{x^{\left [  \frac{\varphi^k}{\sqrt{5}}\right ]}}{1+...}}}}}

\varphi =\frac{1+\sqrt{5}}{2}

A279586



Recurrences (Pisot and related sequences)



a(n) = floor(a(n-1)^2/a(n-2)), a(0) = 3, a(1) = 16.


a(n)=\left [ x^{n} \right ]  \frac{3-2x+x^2-x^3}{1-6x+4x^2-2x^3+2x^4}

A278681


a(n) = floor(a(n-1)^2/a(n-2)), a(0) = 4, a(1) = 14.


a(n)=\left [ x^{n} \right ]  \frac{4-2x+x^2-x^3}{1-4x+2x^2-x^3+x^4}

A278692


a(n) = floor(a(n-1)^2/a(n-2)), a(0) = 5, a(1) = 13.


a(n)=\left [ x^{n} \right ]  \frac{5-2x+4x^2-5x^3+x^4-2x^5}{(1-x)(1-2x-3x^3-x^5)}

A278764


a(n) = ceiling(a(n-1)^2/a(n-2)), a(0) = 4, a(1) = 14.


a(n)=\left [ x^{n} \right ]  \frac{4+2x-x^2-3x^3-2x^4-2x^5+2x^6-x^7}{(1-x)(1-2x-4x^2-4x^3-2x^4-x^5+x^6-x^7)}

A277084


a(n) = ceiling(a(n-1)^2/a(n-2)), a(0) = 5, a(1) = 12.


a(n)=\left [ x^{n} \right ]  \frac{5-3x+3x^2-2x^3+x^5-3x^6-x^7-2x^8}{(1-x)(1-2x-2x^3-x^4-x^5-2x^6-x^7-x^8)}

A277088


a(n) = ceiling(a(n-1)^2/a(n-2)), a(0) = 6, a(1) = 15.


a(n)=\left [ x^{n} \right ]  \frac{6-3x-x^2-2x^3+x^4+3x^5-5x^6}{(1-x)(1-2x-x^2-x^3-2x^6)}

A277089



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