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User:Ilya Gutkovskiy

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

Generalization of generating functions

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The ordinary generating function for the generalized k-gonal numbers

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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, ...

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

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, ...

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

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, ...

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

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, ...

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

The ordinary generating function for the integers repeated k times

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

Double hyperfactorial

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

Conjectures

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

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).

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

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

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

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

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

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

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

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

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).

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

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

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

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).

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

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.

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

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.

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

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.

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

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

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

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.

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

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.

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

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

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

Recurrences (Pisot and related sequences)

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

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

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

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

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

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

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

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

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

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

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

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