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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 k-4 generalized k-gonal numbers (for k >= 8).

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) = 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) = 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.

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